Analog charged black hole formation via percolation: Exploring cosmic censorship and Hoop conjecture

Nitesh Jaiswal; S. Shankaranarayanan (IIT Bombay)

REVIEW 4 major objections 5 minor 111 references

Analog charged black hole formation via percolation: Exploring cosmic censorship and Hoop conjecture

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A chiral spin chain's percolation clusters reproduce charged-black-hole critical exponents.

desk verdict The chiral spin-chain mapping is sound, but the central exponent correspondence collapses: the surface-gravity fits use horizonless spacetimes and the 'percolation' never defines a probability. read the letter →

arxiv 2502.09317 v2 pith:2XB2E3YW submitted 2025-02-13 gr-qc cond-mat.stat-mechcond-mat.str-elhep-th

Nitesh Jaiswal , S. Shankaranarayanan (IIT Bombay) This is my paper · ORCID

classification gr-qccond-mat.stat-mechcond-mat.str-elhep-th MSC 83C5782B4383C75 PACS 04.70.-s05.50.+q64.60.ah

keywords analogblackholespercolationtheorychiralspinchainhoopconjecturecosmiccensorshipcriticalexponentsReissner-NordströmFock-spacegraph

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that a one-dimensional chiral spin chain, read as a classical bond-percolation model on its Fock-space graph, reproduces the critical behaviour of charged black hole formation. In the phase where next-nearest-neighbour chiral interactions dominate, the correlation-length exponent $\nu_P = 1.11$ from cluster growth matches the surface-gravity exponent $\gamma = 1.00$, and the total-energy exponent $\beta = 0.53$ matches the black-hole mass exponent $\gamma_M = 0.5$ of the continuum analog metric. The authors conclude that exponential growth of cluster size and energy, not the geometric hoop condition, is the key indicator of horizon formation, and they read the absence of horizons for $\sigma > \sigma_c$ as analog cosmic censorship. If the correspondence holds, a deterministic spin-chain system can serve as a laboratory for black-hole critical phenomena and provides a new formation criterion for analogue horizons.

What carries the argument

The load-bearing object is the Fock-space graph of the mean-field chiral spin chain, treated as a deterministic bond-percolation lattice. Nodes are fermionic Fock states; edges are non-zero matrix elements of the Hamiltonian, active for nearest-neighbour hopping when $U$ dominates and for next-nearest-neighbour hopping when $v$ dominates. The chirality operator $\chi_j = \vec{\sigma}_j \cdot (\vec{\sigma}_{j+1} \times \vec{\sigma}_{j+2})$ creates the long-range three-spin coupling that defines the NNN-dominated black-hole phase. From this graph the paper extracts $\nu_P$ from $\log S$ versus $\log 2^N$ and $\beta$ from $\log E_T$ versus $\log 2^N$, and compares them with exponents computed analytically from the Gullstrand-Painlevé continuum metric.

What would settle it

Compute the largest-cluster size $S$ and total energy $E_T$ in the NNN-dominated phase at substantially larger $N$ (say $N = 30$–$40$): if $\log S$ versus $\log 2^N$ departs from slope $1.11$, or if the exponential growth of clusters and energy saturates, Table I's correspondence and the sufficiency claim collapse. Alternatively, introduce a random-bond version of the graph with occupation probability $P$, locate $P_c$, and check whether the resulting exponents match $\nu_P = 1.11$ and $\beta = 0.53$; different exponents would show the analogy is an artifact of the deterministic activation rule.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a correspondence between two scaling laws. In the NNN-dominated phase of the chiral spin chain, the largest Fock-space cluster grows as $S \sim 2^N$ with $\nu_P = 1.11$ and the total energy as $E_T \sim (2^N)^{\beta/\nu_P}$ with $\beta = 0.53$; the same model in the continuum limit yields a Gullstrand-Painlevé metric with two horizons, surface gravity $\kappa_g \sim (x - x_c)^{1.00}$, and Komar mass $M_{\rm BH} \sim (\sigma - \sigma_c)^{0.5}$. Table I equates these pairs, identifying percolation cluster growth with the critical collapse of a charged black hole. The paper further claims that the hoop conjecture is satisfied in both the NN- and NNN-dominated phases and therefore cannot select the black-hole phase; exponential cluster and energy growth is the distinguishing indicator, and the absence of horizons for $\sigma > \sigma_c$, hence of naked singularities, is presented as cosmic censorship in the analog.

Load-bearing premise

The correspondence rests on treating the deterministic Fock-space graph as a genuine percolation system with a critical point, even though the paper never defines a bond-occupation probability $P$ or a critical value $P_c$; the exponents come from finite-size log-log fits in $N$.

Editorial extensions

If this is right

An experimental spin-chain or optical-lattice realization could probe charged-black-hole critical exponents without active feedback, because bond activation is fixed by the Hamiltonian couplings.
The hoop conjecture is necessary but not sufficient: geometry alone cannot decide horizon formation, and exponential growth of cluster size and energy becomes the operative criterion.
The analog model supports cosmic censorship in the sense that supercritical charge ($\sigma > \sigma_c$) yields a regular spacetime with no horizon and no naked singularity.
Larger finite-size simulations should sharpen the match between lattice percolation exponents and continuum gravitational exponents, since the paper finds agreement already at $N \le 20$.
A quantum version of the model, replacing classical clusters with entangled states, could connect percolation entanglement entropy to black-hole entropy and bear on the information paradox.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The reported exponents do not match ordinary percolation universality classes (mean-field $\nu = 1/2$, $\beta = 1$ in high dimensions; $\nu = 4/3$ in two dimensions), so if they hold they define a universality class peculiar to this deterministic Fock-space graph rather than standard random percolation.
A sharper test would derive $\nu_P$ and $\beta$ analytically from the Hamiltonian couplings instead of finite-size log-log fits, and would check whether the correspondence survives when a genuine bond occupation probability $P$ is introduced.
If the exponential-growth criterion generalizes, horizon formation in analogue systems could be diagnosed by the scaling of configuration-space connectivity rather than by metric geometry alone.
The $\sigma \leftrightarrow Q/M$ analogy suggests the model can probe near-extremal charged black holes at $\sigma \approx \sigma_c$, where surface gravity vanishes and Hawking evaporation slows.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things you should know about arXiv:2502.09317. First, the chiral spin-chain to analog metric construction is legitimate and follows directly from cited prior work. The observation that cluster growth is linear in the NN-dominated phase and exponential in the NNN-dominated phase is also a genuine numerical fact. Second, the central claim of Table I, that percolation exponents ν_P=1.11 and β=0.53 match gravitational exponents γ=1.00 and γ_M=0.5, is not supported by the paper's own equations.

The stress-test note is right about the gravitational half. With U(x)=sech(x/σ)/sqrt(2σ), g(x)=1−(σ/σ_c)cosh²(x/σ). At the horizon g(x_c)=0, Eq. (14) gives κ²_g as a nonzero constant, not κ² ∝ (x−x_c)². Worse, the two fits that produce γ=1.00 use v=0.1, σ=50.1 and v=0.5, σ=2.1. For both, σ>σ_c=1/(2v²), so there is no horizon and x_c is complex. The exponent γ=1.00 is therefore a fit to a nonexistent horizon. It is not a consequence of the metric they printed. That single problem breaks the correspondence.

The percolation side is equally shaky. Eq. (19) assumes ξ∼|P−P_c|^{−ν_P}∼2^N, but nowhere do the authors define a bond occupation probability P or a critical probability P_c. Bond activation is fixed by whether U or v dominates; there is no randomness and no threshold. The exponents are slopes of small-N log-log fits without error bars. The choice of U(x)=sech is made by hand to produce two horizons, and the matching of exponents is post hoc.

I want to give credit where it is due. The low-energy metric derivation is competently done, the connection to the chiral spin-chain literature is honest, and the exponential-vs-linear growth distinction could be useful as a diagnostic for analog horizon formation. The hoop conjecture discussion is interesting, but it rests on the same unsupported identification: the percolation cluster is never shown to be a genuine percolation critical system.

These are not stylistic weaknesses. They are load-bearing. The paper is not coherent on its own terms: it presents fits to a horizon that its own parameters rule out. I would not send this to a serious referee in its current form. The authors would need to redo the gravity-side scaling for σ<σ_c, define a real percolation threshold with uncertainties, and provide an independent prediction before the claim of correspondence can be taken seriously. As it stands, it is a desk reject.

Referee Report

4 major / 5 minor

Summary. The paper proposes an analog model of charged black hole formation built from a one-dimensional chiral spin chain. A Jordan-Wigner transformation and continuum limit yield an effective Dirac field on a Gullstrand-Painlevé-type metric, and a specific profile U(x)=sech(x/σ)/√(2σ) is chosen so that the metric has two Killing horizons, resembling Reissner-Nordström. The authors compute surface gravity and Komar mass and compare their scaling exponents with exponents extracted from a Fock-space 'classical percolation' model of the same spin chain. They report agreement between ν_P=1.11 and γ=1.00 and between β=0.53 and γ_M=0.5, and they conclude that exponential growth of clusters and energy is the key indicator of black hole formation while the hoop conjecture is necessary but not sufficient.

Significance. If the claimed correspondence were valid, the paper would offer an interesting bridge between Fock-space connectivity in a deterministic spin-chain model and critical exponents of black hole formation, with potential implications for analog gravity and cosmic censorship. The derivation of the emergent Gullstrand-Painlevé metric from the chiral spin chain is systematic, and the observation that NNN-dominated Fock-space connectivity grows exponentially while NN-dominated connectivity grows only linearly is concrete and potentially useful. However, the central numerical correspondence in Table I is not supported by the manuscript's own equations: the reported surface-gravity exponent is obtained for parameter values where no real horizon exists, and the percolation exponents are extracted without a defined bond-occupation probability or critical probability. The significance of the paper therefore rests on claims that, as written, cannot be verified.

major comments (4)

[Section III and Appendix C, Eq. (14)/(C3), Fig. 3(right), Table I] The reported surface-gravity exponent γ=1.00 is not a property of the printed metric. With U(x)=sech(x/σ)/√(2σ), the horizon condition g(x)=0 gives x_c=±σ cosh^{-1}(√(σ_c/σ)) only for σ<σ_c. Evaluating Eq. (C3) at g=0 yields κ_g² = [3/(4σ σ_c²)](σ_c/σ − 1), a positive constant, not a power law in (x−x_c). The fits in Fig. 3 use v=0.1, σ=50.1 and v=0.5, σ=2.1, both with σ>σ_c=1/(2v²); for these values x_c is complex and no real horizon exists. The claimed γ=1.00 is therefore a fit to a nonexistent horizon, and the first numerical pillar of Table I is unsupported.
[Appendix C, Eq. (16), Fig. 4(right), Table I] The same problem affects the Komar-mass exponent γ_M=0.5. Equation (16) evaluates the boundary contribution at a real inner horizon, which requires σ<σ_c. The fits in Fig. 4 use v=0.1, σ=50.1 and v=0.5, σ=2.1, both with σ>σ_c, so the argument of cosh^{-1} in Eq. (16) is less than one and the horizon is absent. The reported scaling MBH ∝ (σ−σ_c)^{0.5} is therefore not supported for the parameter values used in the numerical comparison.
[Section IV A, Eq. (19)] The percolation exponents are not defined in the manuscript. No bond-occupation probability P or critical probability P_c is introduced; bond activation is deterministic, depending on whether U or v dominates. Equation (19) asserts ξ ∼ |P−P_c|^{−ν_P} ∼ 2^N without specifying P(N), P_c, or the relation between the correlation length and the system size. Consequently ν_P=1.11 and β=0.53 are slopes of log-log fits for N≤20 with no error bars, no disorder average, and no finite-size scaling analysis. Calling them universal percolation critical exponents is not justified.
[Section V and Table I] The claimed correspondence between the percolation exponents and the gravitational exponents is post hoc. The metric profile U(x)=sech(x/σ)/√(2σ) is chosen by hand to produce two horizons, and the gravitational exponents are computed from this chosen metric; the percolation exponents are fitted from a separate, deterministic Fock-space construction. No derivation connects ν_P to γ or β to γ_M. With γ=1.00 already invalidated by the horizonless parameter values, the numerical agreement in Table I cannot support the paper's central conclusion, and the subsequent claim that exponential growth is the key indicator of black hole formation lacks a quantitative foundation.

minor comments (5)

[Section IV A] The symbol N is used both for the number of lattice sites and for the number of clusters in the percolation analysis, which makes the description of Fig. 2 difficult to follow.
[Eq. (6) and Appendix A, Eq. (A9)] The dispersion relation in Eq. (6) is written as E(p)=ũ(p)+ũ*(p)+v sin p, while Appendix A gives E(p)=−2U cos p + v sin(2p). The relation between these two expressions should be stated explicitly.
[Fig. 3 caption and Section IV A] The caption states that the filled circles are 'analytical values using Eq. (14)', but the text reports fits for σ>σ_c where Eq. (14) is evaluated for a complex horizon. The actual x-range and parameter values used in the plot should be specified.
[Eq. (18)] The definition of energy density as '# of connected bonds / # of connected states × IE' is ambiguous for clusters that contain both NN and NNN bonds, since IE is taken to be either U or v; the rule for mixed clusters should be stated.
[Appendix D] The hoop-conjecture analysis identifies the smallest cluster's number of states with circumference C and the number of connected bonds with mass M, but no quantitative mapping to the gravitational hoop condition C ≤ 4πM is provided, so the conclusion that the hoop conjecture is necessary but not sufficient remains schematic.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: percolation exponents and gravitational exponents are obtained independently and only compared post hoc.

full rationale

The claimed correspondence is a post-hoc comparison of two independently obtained sets of exponents. The percolation exponents ν_P=1.11 and β=0.53 are extracted from numerical fits to the Fock-space cluster data of the deterministic spin-chain graph (Sec. IV A, Figs. 3-4). The gravitational exponents γ=1.00 and γ_M=0.5 are obtained from the analytic metric with U(x)=sech(x/σ)/√(2σ): γ by fitting Eq. (14), γ_M by fitting Eq. (16) (Appendix C). Neither set is defined in terms of the other; no equation is substituted into itself, and no fitted parameter is relabeled as a prediction. The metric-to-percolation mapping is imported from Refs. [68,69,80] (Pachos and coauthors), not from the present authors, and the self-citations ([81], [84], [102]) are contextual or standard-coordinate references rather than load-bearing steps. The absence of a defined bond probability P and the apparent use of σ>σ_c where Eq. (12) predicts no real horizon are serious correctness/validity concerns, but they are not circularity: the exponents would fail independently of any self-referential construction. Hence no circular step can be exhibited.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central correspondence rests on a hand-chosen metric profile, hand-picked near-critical parameters, fitted exponents, and the assumption that a deterministic Fock-space graph behaves as a percolation model. No new physical entities are introduced; the horizon-like boundary is an interpretation of NNN-dominated regions, not an independent entity.

free parameters (5)

chosen metric profile U(x) = sech(x/sigma)/sqrt(2 sigma) = sech(x/sigma)/sqrt(2 sigma)
Introduced by hand in Section III, Eq. (11), to make g(x) = 1 - v^2/U^2(x) develop two horizons. It is not derived from the spin-chain couplings.
chiral coupling v = 0.1, 0.5 in the scaling plots
Dimensionless Hamiltonian coupling that sets sigma_c = 1/(2 v^2). Values are chosen near criticality rather than fixed by data.
profile width sigma = 50.1, 2.1 in the scaling plots
Controls the analog charge-to-mass ratio sigma/sigma_c. The values used for surface gravity scaling are above sigma_c, where no horizon exists.
percolation exponents nu_P and beta = nu_P = 1.11, beta = 0.53
Obtained from slopes of log-log fits in Fig. 3(left) and Fig. 4(left) for N <= 20, with no reported uncertainties. The central correspondence in Table I depends on these fitted values.
gravity-side exponents gamma and gamma_M = gamma = 1.00, gamma_M = 0.50
Derived by fitting Eq. (14) and Eq. (16) respectively. These are exponents of the chosen sech metric, not independent predictions.

assumptions (5)

domain assumption Mean-field decoupling of the Jordan-Wigner fermionized Hamiltonian is valid and captures the low-energy continuum behavior.
Used in Section II and Appendix A to obtain H_MF; no error estimate or exact diagonalization check is provided for the analog metric.
ad hoc to paper The Fock-space graph with edges weighted by Hamiltonian matrix elements is a valid classical percolation model with a critical probability P_c.
Section IV constructs a deterministic graph and invokes xi ~ |P - P_c|^{-nu} ~ 2^N without defining a random bond occupation probability or a threshold P_c.
domain assumption Near the percolation threshold, the correlation length scales as 2^N, so log-log fits in N yield critical exponents.
Eq. (19) imports standard percolation scaling, but it is not justified for this deterministic Fock-space graph.
ad hoc to paper The number of states in the smallest cluster and the number of connected bonds represent hoop circumference C and mass M.
Appendix D relies entirely on this identification to draw the hoop-conjecture conclusion.
ad hoc to paper Surface gravity and Killing horizon formulas apply even when sigma > sigma_c, where g(x) = 0 has no real solution.
Appendix C and Fig. 3(right) evaluate kappa_g for sigma = 50.1 and sigma = 2.1, both above sigma_c = 50 and sigma_c = 2 respectively.

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Pith. "Pith review of Analog charged black hole formation via percolation: Exploring cosmic censorship and Hoop conjecture." pith.science (2026). https://pith.science/paper/2XB2E3YW

@misc{pith2026250209317,
  author       = {Pith},
  title        = {Pith review of: Analog charged black hole formation via percolation: Exploring cosmic censorship and Hoop conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2XB2E3YW}},
  note         = {Machine review of arXiv:2502.09317}
}

read the original abstract

We investigate an analog model of charged black hole (BH) formation using the framework of classical percolation. By analyzing the scaling behavior of key quantities, including surface gravity and Komar mass, we establish a robust correspondence between this analog system and gravitational collapse in general relativity. Our numerical simulations of the lattice model show excellent agreement with analytical predictions for the continuum limit, highlighting the potential of analog systems to capture essential features of BH physics. Interestingly, we find that while geometric criteria related to the hoop conjecture are necessary, they are not sufficient for BH formation in our model. Instead, the exponential growth of energy and cluster size emerges as the key indicator, suggesting a novel interpretation of the hoop conjecture and providing further support for cosmic censorship within our analog framework by ensuring horizon formation. This work offers a fresh perspective on the organization of matter within BH event horizons and lays the groundwork for future quantum extensions that could shed light on Hawking radiation and the BH information paradox by linking entanglement entropy in quantum percolation models to BH entropy.

Figures

Figures reproduced from arXiv: 2502.09317 by the authors.

**Figure 2.** FIG. 2: Normalized cluster energy density (left) and [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** FIG. 3: Plot of log [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4: Plot of log [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

**Figure 5.** Figure 5: shows the number of connected bonds Eb, and the number of states C in the smallest cluster. The equal slopes of the plotted lines for both the NN- and NNN-dominated phases suggest that the hoop conjecture [71–74], should be satisfied, requiring U ≥ 1 in the NN phase an…

**Figure 6.** Figure 6: FIG. 6: Inverse fourth root of the Kretschmann scalar, [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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Interesting limits of the BH metric Let us now explore the interesting limits of the metric Eq. (B9). a. Carrollian limit Setting v = 0 in the above metric, the dispersion relation near the Fermi pointp0 becomes E(p + p0) = ± U(x) p (π − acp)2. In the limit of U (x) → 0, the momentum-space dispersion relation flattens completely, corresponding to the Carr...

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Later in this section while performing the Fourier transformation, we will show that V (j, j+ 2) only modifies the overall energy scale of the system

− j|), to depend only on the distance between the sites so that the system remains translationally invariant. Later in this section while performing the Fourier transformation, we will show that V (j, j+ 2) only modifies the overall energy scale of the system. In the continuum limit, the system is described by a massless, minimally coupled Dirac field in ...

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Defined as K = RµαβρRµαβρ, the Kretschmann scalar, unlike the Ricci scalar, remains non-zero in vacuum regions, providing valuable information where the Ricci scalar is trivial

Kretschmann scalar and surface gravity In our study, we focus on the Kretschmann scalar (K), a quadratic scalar invariant having dimensions [ L]−4. Defined as K = RµαβρRµαβρ, the Kretschmann scalar, unlike the Ricci scalar, remains non-zero in vacuum regions, providing valuable information where the Ricci scalar is trivial. For the metric (10), we find th...

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Komar mass To make the connections between the percolation model and gravitational collapse more concrete, we now turn to the Komar mass, which represents the total mass (or energy) contained within the spacetime, as seen from infinity. The Komar mass MK for (1 + 1)−D space-time in terms of Killing vector K ν is given by [88], MK = Z S dx p |g|nµRµνKν , (...

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